On the Existence, Uniqueness, and Stability of Periodic Modes of Motion of a Locomotion System with a Mobile Internal Mass

In this paper, we consider the rectilinear motion of a locomotion system consisting of a hull and an internal mass in a resisting medium during periodic movement of the internal mass relative to the hull. The periodic modes of the system’s movement, in which the hull’s speed is also a time periodic function, are studied. The questions of the existence and uniqueness of periodic modes of system motion, their stability with respect to the initial conditions, and the rate of convergence of arbitrary movements in relation to them are studied. The periodic mode of motion of the locomotion system is shown to exist, be unique, and be exponentially stable if the medium resistance is monotonic and unlimitedly increases with increasing speed and if the speed of the internal mass relative to the hull is continuous. A two-sided assessment of the hull’s speed in a periodic mode of motion is obtained. In particular cases of linear and piecewise linear medium resistance, a periodic mode of the system’s motion is constructed, and the rate of exponential convergence is calculated.